### 7.13 Re­flec­tion and Trans­mis­sion Co­ef­fi­cients

Scat­ter­ing and tun­nel­ing can be de­scribed in terms of so-called re­flec­tion and trans­mis­sion co­ef­fi­cients. This sec­tion ex­plains the un­der­ly­ing ideas.

Con­sider an ar­bi­trary scat­ter­ing po­ten­tial like the one in fig­ure 7.22. To the far left and right, it is as­sumed that the po­ten­tial as­sumes a con­stant value. In such re­gions the en­ergy eigen­func­tions take the form

where is the clas­si­cal mo­men­tum and and are con­stants. When eigen­func­tions of slightly dif­fer­ent en­er­gies are com­bined to­gether, the terms pro­duce wave pack­ets that move for­wards in , graph­i­cally from left to right, and the terms pro­duce pack­ets that move back­wards. So the sub­scripts in­di­cate the di­rec­tion of mo­tion.

This sec­tion is con­cerned with a sin­gle wave packet that comes in from the far left and is scat­tered by the non­triv­ial po­ten­tial in the cen­ter re­gion. To de­scribe this, the co­ef­fi­cient must be zero in the far-right re­gion. If it was nonzero, it would pro­duce a sec­ond wave packet com­ing in from the far right.

In the far-left re­gion, the co­ef­fi­cient is nor­mally not zero. In fact, the term pro­duces the part of the in­com­ing wave packet that is re­flected back to­wards the far left. The rel­a­tive amount of the in­com­ing wave packet that is re­flected back is called the “re­flec­tion co­ef­fi­cient” . It gives the prob­a­bil­ity that the par­ti­cle can be found to the left of the scat­ter­ing re­gion af­ter the in­ter­ac­tion with the scat­ter­ing po­ten­tial. It can be com­puted from the co­ef­fi­cients of the en­ergy eigen­func­tion in the left re­gion as, {A.32},

 (7.72)

Sim­i­larly, the rel­a­tive frac­tion of the wave packet that passes through the scat­ter­ing re­gion is called the “trans­mis­sion co­ef­fi­cient” . It gives the prob­a­bil­ity that the par­ti­cle can be found at the other side of the scat­ter­ing re­gion af­ter­wards. It is most sim­ply com­puted as : what­ever is not re­flected must pass through. Al­ter­na­tively, it can be com­puted as

 (7.73)

where re­spec­tively are the val­ues of the clas­si­cal mo­men­tum in the far left and right re­gions.

Note that a co­her­ent wave packet re­quires a small amount of un­cer­tainty in en­ergy. Us­ing the eigen­func­tion at the nom­i­nal value of en­ergy in the above ex­pres­sions for the re­flec­tion and trans­mis­sion co­ef­fi­cients will in­volve a small er­ror. It can be made to go to zero by re­duc­ing the un­cer­tainty in en­ergy, but then the size of the wave packet will ex­pand cor­re­spond­ingly.

In the case of tun­nel­ing through a high and wide bar­rier, the WKB ap­prox­i­ma­tion may be used to de­rive a sim­pli­fied ex­pres­sion for the trans­mis­sion co­ef­fi­cient, {A.29}. It is

 (7.74)

where and are the turn­ing points in fig­ure 7.22, in be­tween which the po­ten­tial en­ergy ex­ceeds the to­tal en­ergy of the par­ti­cle.

There­fore in the WKB ap­prox­i­ma­tion, it is just a mat­ter of do­ing a sim­ple in­te­gral to es­ti­mate what is the prob­a­bil­ity for a wave packet to pass through a bar­rier. One fa­mous ap­pli­ca­tion of that re­sult is for the al­pha de­cay of atomic nu­clei. In such de­cay a so-called al­pha par­ti­cle tun­nels out of the nu­cleus.

For sim­i­lar con­sid­er­a­tions in three-di­men­sion­al scat­ter­ing, see ad­den­dum {A.30}.

Key Points
A trans­mis­sion co­ef­fi­cient gives the prob­a­bil­ity for a par­ti­cle to pass through an ob­sta­cle. A re­flec­tion co­ef­fi­cient gives the prob­a­bil­ity for it to be re­flected.

A very sim­ple ex­pres­sion for these co­ef­fi­cients can be ob­tained in the WKB ap­prox­i­ma­tion.